Integrand size = 29, antiderivative size = 382 \[ \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{2 a x^2}+\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a \sqrt {d}}+\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2 \sqrt {d}}-\frac {\sqrt {c} \left (b^2 d-2 a c d-a b e+\sqrt {b^2-4 a c} (b d-a e)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {c} \left (b^2 d-b \left (\sqrt {b^2-4 a c} d+a e\right )-a \left (2 c d-\sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
1/2*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))/a/d^(1/2)+(-a*e+b*d)*arctanh((e*x^2 +d)^(1/2)/d^(1/2))/a^2/d^(1/2)-1/2*(e*x^2+d)^(1/2)/a/x^2-1/2*arctanh(2^(1/ 2)*c^(1/2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2) *(b^2*d-2*a*c*d-a*b*e+(-a*e+b*d)*(-4*a*c+b^2)^(1/2))/a^2*2^(1/2)/(-4*a*c+b ^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+1/2*arctanh(2^(1/2)*c^(1/ 2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(b^2*d- 2*a*c*d-a*b*e-(-a*e+b*d)*(-4*a*c+b^2)^(1/2))/a^2*2^(1/2)/(-4*a*c+b^2)^(1/2 )/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 1.05 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\frac {-\frac {a \sqrt {d+e x^2}}{x^2}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d-2 a c d+b \sqrt {b^2-4 a c} d-a b e-a \sqrt {b^2-4 a c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (-b^2 d+2 a c d+b \sqrt {b^2-4 a c} d+a b e-a \sqrt {b^2-4 a c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {(2 b d-a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}}}{2 a^2} \]
(-((a*Sqrt[d + e*x^2])/x^2) + (Sqrt[2]*Sqrt[c]*(b^2*d - 2*a*c*d + b*Sqrt[b ^2 - 4*a*c]*d - a*b*e - a*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqr t[d + e*x^2])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c ]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 2*a*c*d + b*Sqrt[b^2 - 4*a*c]*d + a*b*e - a*Sqrt[b^2 - 4*a*c]*e)*ArcTan[( Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]] )/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]) + ((2*b*d - a*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/Sqrt[d])/(2*a^2)
Time = 2.38 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1578, 1199, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {e x^2+d}}{x^4 \left (c x^4+b x^2+a\right )}dx^2\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {\int \left (\frac {d e^2}{a \left (d-x^4\right )^2}+\frac {(b d-a e) e}{a^2 \left (d-x^4\right )}-\frac {\left (b \left (c d^2-b e d+a e^2\right )-c (b d-a e) x^4\right ) e}{a^2 \left (c x^8-(2 c d-b e) x^4+c d^2+a e^2-b d e\right )}\right )d\sqrt {e x^2+d}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt {c} e \left (\sqrt {b^2-4 a c} (b d-a e)-a b e-2 a c d+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {c} e \left (-\sqrt {b^2-4 a c} (b d-a e)-a b e-2 a c d+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {e (b d-a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2 \sqrt {d}}+\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a \sqrt {d}}+\frac {e^2 \sqrt {d+e x^2}}{2 a \left (d-x^4\right )}}{e}\) |
((e^2*Sqrt[d + e*x^2])/(2*a*(d - x^4)) + (e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt [d]])/(2*a*Sqrt[d]) + (e*(b*d - a*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(a^ 2*Sqrt[d]) - (Sqrt[c]*e*(b^2*d - 2*a*c*d - a*b*e + Sqrt[b^2 - 4*a*c]*(b*d - a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b ^2 - 4*a*c])*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^ 2 - 4*a*c])*e]) + (Sqrt[c]*e*(b^2*d - 2*a*c*d - a*b*e - Sqrt[b^2 - 4*a*c]* (b*d - a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + S qrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sq rt[b^2 - 4*a*c])*e]))/e
3.4.59.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Time = 0.82 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}}{2 a \,x^{2}}-\frac {-\frac {\left (-a e +2 b d \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{a \sqrt {d}}+\frac {c \sqrt {2}\, \left (-\frac {\left (a b \,e^{2}+2 a c d e -b^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-a b \,e^{2}-2 a c d e +b^{2} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}}{2 a}\) | \(364\) |
| pseudoelliptic | \(\frac {\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c \,x^{2} \left (\frac {\left (e a \sqrt {d}-d^{\frac {3}{2}} b \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (\left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}+\frac {b e a \sqrt {d}}{2}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\sqrt {2}\, c \,x^{2} \left (\frac {\left (-e a \sqrt {d}+d^{\frac {3}{2}} b \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (\left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}+\frac {b e a \sqrt {d}}{2}\right )\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\frac {\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (x^{2} \left (a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{\sqrt {d}}\right )+\sqrt {e \,x^{2}+d}\, \sqrt {d}\, a \right )}{2}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, a^{2} x^{2}}\) | \(446\) |
| default | \(\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {e \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{2 d}}{a}-\frac {b \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{a^{2}}-\frac {-\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c \left (\left (a e -b d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (a b e +2 a c d -b^{2} d \right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\sqrt {2}\, c \left (\left (a e -b d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-e \left (a b e +2 a c d -b^{2} d \right )\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-2 \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, b \sqrt {e \,x^{2}+d}\right )}{2 a^{2} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(493\) |
-1/2*(e*x^2+d)^(1/2)/a/x^2-1/2/a*(-(-a*e+2*b*d)/a/d^(1/2)*ln((2*d+2*d^(1/2 )*(e*x^2+d)^(1/2))/x)+1/a*c*2^(1/2)/(-e^2*(4*a*c-b^2))^(1/2)*(-(a*b*e^2+2* a*c*d*e-b^2*d*e+(-e^2*(4*a*c-b^2))^(1/2)*a*e-(-e^2*(4*a*c-b^2))^(1/2)*b*d) /((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x^2+d)^(1/2) *2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))+(-a*b*e^2-2*a*c* d*e+b^2*d*e+(-e^2*(4*a*c-b^2))^(1/2)*a*e-(-e^2*(4*a*c-b^2))^(1/2)*b*d)/((b *e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x^2+d)^(1/2)*2^(1/ 2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 3290 vs. \(2 (309) = 618\).
Time = 260.18 (sec) , antiderivative size = 6592, normalized size of antiderivative = 17.26 \[ \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {d + e x^{2}}}{x^{3} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (309) = 618\).
Time = 0.33 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=-\frac {{\left (2 \, b d - a e\right )} \arctan \left (\frac {\sqrt {e x^{2} + d}}{\sqrt {-d}}\right )}{2 \, a^{2} \sqrt {-d}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} e^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c d^{2} - \sqrt {b^{2} - 4 \, a c} b^{2} d e + \sqrt {b^{2} - 4 \, a c} a b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | e \right |} + {\left (b^{3} d e^{2} - a b^{2} e^{3} - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e + \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b d e + \sqrt {b^{2} - 4 \, a c} a^{3} e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} e^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c d^{2} - \sqrt {b^{2} - 4 \, a c} b^{2} d e + \sqrt {b^{2} - 4 \, a c} a b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | e \right |} + {\left (b^{3} d e^{2} - a b^{2} e^{3} - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e - \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b d e + \sqrt {b^{2} - 4 \, a c} a^{3} e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {\sqrt {e x^{2} + d}}{2 \, a x^{2}} \]
-1/2*(2*b*d - a*e)*arctan(sqrt(e*x^2 + d)/sqrt(-d))/(a^2*sqrt(-d)) + 1/8*( sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^3 - 4*a*b*c)*d - (a*b ^2 - 4*a^2*c)*e)*e^2 - 2*(sqrt(b^2 - 4*a*c)*b*c*d^2 - sqrt(b^2 - 4*a*c)*b^ 2*d*e + sqrt(b^2 - 4*a*c)*a*b*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a *c)*c)*e)*abs(e) + (b^3*d*e^2 - a*b^2*e^3 - 2*(b^2*c - 2*a*c^2)*d^2*e)*sqr t(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x ^2 + d)/sqrt(-(2*a^2*c*d - a^2*b*e + sqrt(-4*(a^2*c*d^2 - a^2*b*d*e + a^3* e^2)*a^2*c + (2*a^2*c*d - a^2*b*e)^2))/(a^2*c)))/((sqrt(b^2 - 4*a*c)*a^2*c *d^2 - sqrt(b^2 - 4*a*c)*a^2*b*d*e + sqrt(b^2 - 4*a*c)*a^3*e^2)*abs(c)*abs (e)) - 1/8*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*((b^3 - 4*a*b *c)*d - (a*b^2 - 4*a^2*c)*e)*e^2 + 2*(sqrt(b^2 - 4*a*c)*b*c*d^2 - sqrt(b^2 - 4*a*c)*b^2*d*e + sqrt(b^2 - 4*a*c)*a*b*e^2)*sqrt(-4*c^2*d + 2*(b*c + sq rt(b^2 - 4*a*c)*c)*e)*abs(e) + (b^3*d*e^2 - a*b^2*e^3 - 2*(b^2*c - 2*a*c^2 )*d^2*e)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1 /2)*sqrt(e*x^2 + d)/sqrt(-(2*a^2*c*d - a^2*b*e - sqrt(-4*(a^2*c*d^2 - a^2* b*d*e + a^3*e^2)*a^2*c + (2*a^2*c*d - a^2*b*e)^2))/(a^2*c)))/((sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*d*e + sqrt(b^2 - 4*a*c)*a^3*e^2 )*abs(c)*abs(e)) - 1/2*sqrt(e*x^2 + d)/(a*x^2)
Time = 10.83 (sec) , antiderivative size = 19959, normalized size of antiderivative = 52.25 \[ \int \frac {\sqrt {d+e x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
(atan((((a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4* e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 1 8*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/(2*a^4) - (((16*a^5*b*c^4*e^12 + 20*a^5*c^5*d*e^11 + a^3*b^5*c^2*e^12 - 8*a^4*b^3* c^3*e^12 + 20*a^4*c^6*d^3*e^9 + 40*a^2*b^3*c^5*d^4*e^8 - 20*a^2*b^4*c^4*d^ 3*e^9 - 27*a^2*b^5*c^3*d^2*e^10 - 20*a^3*b^2*c^5*d^3*e^9 + 84*a^3*b^3*c^4* d^2*e^10 - 8*a*b^5*c^4*d^4*e^8 + 6*a*b^6*c^3*d^3*e^9 + 2*a*b^7*c^2*d^2*e^1 0 - 3*a^2*b^6*c^2*d*e^11 - 32*a^3*b*c^6*d^4*e^8 + 28*a^3*b^4*c^3*d*e^11 - 36*a^4*b*c^5*d^2*e^10 - 68*a^4*b^2*c^4*d*e^11)/a^4 - ((a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(240*a^6*b*c^4*e^11 + 64*a^6*c^5*d*e^10 + 20*a^4*b^5*c^2*e^1 1 - 140*a^5*b^3*c^3*e^11 + 160*a^5*c^6*d^3*e^8 - 32*a^2*b^6*c^3*d^3*e^8 + 32*a^2*b^7*c^2*d^2*e^9 + 224*a^3*b^4*c^4*d^3*e^8 - 208*a^3*b^5*c^3*d^2*e^9 - 432*a^4*b^2*c^5*d^3*e^8 + 272*a^4*b^3*c^4*d^2*e^9 - 48*a^3*b^6*c^2*d*e^ 10 + 348*a^4*b^4*c^3*d*e^10 + 224*a^5*b*c^5*d^2*e^9 - 648*a^5*b^2*c^4*d*e^ 10))/(2*a^4) - ((a*e - 2*b*d)*((128*a^8*c^4*e^11 + 8*a^6*b^4*c^2*e^11 - 64 *a^7*b^2*c^3*e^11 + 128*a^7*c^5*d^2*e^9 + 32*a^5*b^3*c^4*d^3*e^8 - 24*a^5* b^4*c^3*d^2*e^9 + 64*a^6*b^2*c^4*d^2*e^9 - 256*a^7*b*c^4*d*e^10 - 8*a^5*b^ 5*c^2*d*e^10 - 128*a^6*b*c^5*d^3*e^8 + 96*a^6*b^3*c^3*d*e^10)/a^4 - ((d + e*x^2)^(1/2)*(a*e - 2*b*d)*(1024*a^9*c^4*e^10 + 64*a^7*b^4*c^2*e^10 - 512* a^8*b^2*c^3*e^10 + 1536*a^8*c^5*d^2*e^8 + 128*a^6*b^4*c^3*d^2*e^8 - 896...